THEOR. III. PROP. III.
The Times in which the ſame Space is paſt tho
row by unequal Velocities, have the ſame pro
portion to each other as their Velocities contra
rily taken.
row by unequal Velocities, have the ſame pro
portion to each other as their Velocities contra
rily taken.
Let the two unequal Velocities be A the greater, and B the leſſe:
and according to both theſe let a Motion be made thorow the ſame
Space C D. I ſay the Time in which the Velocity A paſſeth the
Space C D, ſhall be to the Time in which the Velocity B paſſeth the
ſaid Space, as the Velocity B to the Velocity A. As A is to B, ſo let
C D be to C E: Then, by the
former Propoſition, the Time in
77[Figure 77]
which the Velocity A paſſeth
C D, ſhall be the ſame with
the Time in which B paſſeth
C E. But the Time in which
the Velocity B paſſeth C E, is
to the Time in which it paſſeth C D, as C E is to C D: Therefore
the Time in which the Velocity A paſſeth C D, is to the Time in which
the Velocity B paſſeth the ſame C D, as C E is to C D; that is, the Ve
locity B is to the Velocity A: Which was to be proved.
and according to both theſe let a Motion be made thorow the ſame
Space C D. I ſay the Time in which the Velocity A paſſeth the
Space C D, ſhall be to the Time in which the Velocity B paſſeth the
ſaid Space, as the Velocity B to the Velocity A. As A is to B, ſo let
C D be to C E: Then, by the
former Propoſition, the Time in
77[Figure 77]which the Velocity A paſſeth
C D, ſhall be the ſame with
the Time in which B paſſeth
C E. But the Time in which
the Velocity B paſſeth C E, is
to the Time in which it paſſeth C D, as C E is to C D: Therefore
the Time in which the Velocity A paſſeth C D, is to the Time in which
the Velocity B paſſeth the ſame C D, as C E is to C D; that is, the Ve
locity B is to the Velocity A: Which was to be proved.
THEOR. IV. PROP. IV.
If two Moveables move with an Equable Mo
tion, but with unequal Velocities, the Spaces
which they paſſe in unequal Times, are to each
other in a proportion compounded of the pro
portion of the Velocities, and of the propor
tion of the Times.
tion, but with unequal Velocities, the Spaces
which they paſſe in unequal Times, are to each
other in a proportion compounded of the pro
portion of the Velocities, and of the propor
tion of the Times.
Let the two Moveables moving with an Equable Motion, be E and
F: And let the proportion of the Velocity of the Moveable E be
to the Velocity of the Moveable F, as A is to B: And let the Time
in which E is moved, be unto the Time in which F is moved, as C is
to D. I ſay the Space paſſed by E, with the Velocity A in the Time C, is to
the Space paſſed by F, with the Velocity B in the Time D, in a proportion
compounded of the proportion of the Velocity A to the Velocity B, and of
F: And let the proportion of the Velocity of the Moveable E be
to the Velocity of the Moveable F, as A is to B: And let the Time
in which E is moved, be unto the Time in which F is moved, as C is
to D. I ſay the Space paſſed by E, with the Velocity A in the Time C, is to
the Space paſſed by F, with the Velocity B in the Time D, in a proportion
compounded of the proportion of the Velocity A to the Velocity B, and of